Mushy gave a good answer to question 1, the others are also perfectly fine answers, I'd just like to add a few things.
The 2 main camps as far as browsers are concerned appear to be Chrome and Maxthon 3. I'm a Maxthon fan (make sure it's the proper 3, not the cut down Cloud version) because Chrome appears to have all sorts of issues that Maxthon simply doesn't. I have a feeling the Chrome users are in the majority though so there must be something good about it. There are also people using all sorts of other browsers, even.. *retch*... *huuuugh*.... Internet Explo... *bloaaaggghhhh* Sorry, didn't manage to say it without puking.
What is very important is to clear your Flash Cache after every update from WGT or every time you change the gear in your bag. Go here to do so, select media5.wgt.com and delete it. You'll have to redo your game settings but it is worth doing regularly. Clearing your browser cache is also a good idea. I don't do that very often as I only use Maxthon to actually play the game, nothing else.
I don't think you'll notice any difference between wired and WiFi. Once the game has loaded the hole and is ready to take the shot everything happens in your computer until the next view is loaded.
Yep, Mushy said it right regarding chipping, pitching and so on. My advice is to find a shot that you can use in most situations and stick with it. I use a punched 64 wedge for every shot between 10 and 41 yards. Under 10 yards I pitch unless I have a bunker between myself and the hole and need the ball to pretty much stop dead upon landing. In that case I'll use a flop but since I very rarely sue it I'm not that good at it.
As far as I know the majority of people find a wired mouse better because the delay between clicking and the click actually being registered is much shorter.
Have a look round the forums, there's tons of good advice. A fair bit of poor advice too but you'll soon learn to filter it a bit.